Let $P$ be a prime ideal of height $n$ in a local ring $R$, which is generated by $n$ elements, say $P=(a_1,...,a_n)$. The image $\bar P$ of $P$ in $R/(a_1,...,a_i)$ , $1≤i≤n$, is an $(n-i)$-generated prime. I want to be sure that it is of height $n-i$.
The height is at most $n-i$, because the height of $\bar P$ is less than or equal to the height of any minimal prime over it in $R/(a_1,...,a_i)$, and the latter is at most $n-i$ by Principal Ideal Theorem. Now, why is it at least $n-i$?
Thanks in advance!